General Physics 1 | Lesson 2.1: Force and Motion Study Notes
General Physics 1: Force and Motion
I. Introduction to Forces
Force Defined: A force is a push or a pull exerted by one object on another.
Vector Quantity: Force is a vector quantity, meaning it possesses both magnitude and direction.
Net Force: The net force is the vector sum of all individual forces acting on a body. Special cases of vector addition are used to find the net force.
A. Types of Forces
Contact Forces: These forces occur when objects are physically touching each other.
Applied Force: A force directly applied by a person or another object.
Drag Force: A resistive force exerted by a fluid (liquid or gas) on an object moving through it.
Frictional Force: A force that opposes motion or impending motion between surfaces in contact.
Spring Force: A restorative force exerted by an elastic spring when it is stretched or compressed.
Normal Force: The component of a contact force perpendicular to the surface that an object rests on or interacts with. It supports the object against gravity or other forces pressing it into the surface.
Noncontact (Field) Forces: These forces act on objects without direct physical contact.
Gravitational Force: The attractive force between any two objects with mass.
Electromagnetic Force: The force between electrically charged particles, encompassing both electric and magnetic forces.
Magnetic Force: Force exerted by magnetic fields on moving electric charges, electric currents, and magnetic materials.
Electrostatic Force: Force between charged particles at rest.
B. Four Fundamental Forces in Nature
These are the forces that govern all interactions in the universe:
Strong Nuclear Force: The strongest force, responsible for holding atomic nuclei together.
Electromagnetic Force: Acts between electrically charged particles.
Weak Nuclear Force: Responsible for radioactive decay.
Gravitational Force: The weakest but longest-ranging force, responsible for planetary orbits and large-scale cosmic structures.
II. Newton's Laws of Motion
Sir Isaac Newton (1643-1727): An English scientist and mathematician renowned for his discovery of the law of gravity and the three laws of motion. He was knighted by Queen Anne in 1705.
Newton's laws describe the motion of objects in our everyday lives.
A. Newton's First Law: Law of Inertia
Statement: An object will remain at rest or continue moving at a constant speed in a straight line (constant velocity) unless acted upon by an external net force.
Inertial Frame of Reference: A frame of reference in which Newton's first law holds true; that is, an object at rest remains at rest, and an object in motion continues to move at constant velocity unless acted upon by a net force. Essentially, it's a non-accelerating reference frame.
Practical Examples:
When a car suddenly stops, your body continues to move forward due to inertia. Seatbelts provide the external force needed to stop you safely.
A kicked soccer ball eventually stops rolling due to external forces like friction and air resistance.
B. Newton's Second Law: Law of Acceleration
Statement: The acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. The direction of the acceleration is the same as the direction of the net force.
Formula: \vec{F}{net} = m\vec{a} or F{net} = ma
Where \vec{F}_{net} is the net force, m is the mass, and \vec{a} is the acceleration.
Practical Examples:
Pushing an empty shopping cart is easier than pushing a full one because the full cart has more mass, requiring more force to achieve the same acceleration.
A baseball accelerates faster when hit with a stronger force (hard swing), demonstrating the direct relationship between force and acceleration.
Mass vs. Weight:
Mass (m):
The amount of matter a body contains.
A scalar quantity.
The mass of a body is constant, regardless of location.
Weight (W):
The measure of the force of gravity exerted by a celestial body (e.g., Earth) on an object.
A vector quantity, always directed toward the center of the gravitating body.
The weight depends on the local value of the acceleration due to gravity (g).
Formula: W = mg
On Earth, approximately g = 9.8 \text{ m/s}^2
On the Moon, approximately g = 1.62 \text{ m/s}^2
C. Newton's Third Law: Law of Interaction
Statement: For every action, there is an equal and opposite reaction.
This means that if object A exerts a force on object B (action force), then object B simultaneously exerts an equal in magnitude and opposite in direction force on object A (reaction force).
Practical Examples:
When you jump off a small boat, you move forward (action) while the boat moves backward (reaction).
Airplanes fly because their engines push air backward (action), and the reaction force (thrust) pushes the plane forward.
Rocket Propulsion: A prime example where the action force is the force generated by the exhaust gas, and the reaction force is the thrust propelling the rocket at high velocity. NASA tested a 3D-printed rocket engine producing approximately 90,000 \text{ N} (or 20,000 \text{ pounds}) of thrust.
D. Apparent Weight in an Elevator
The reading on a weighing scale in an elevator represents the normal force exerted by the scale on the person, which is their apparent weight.
Let m be the mass of the person and g be the acceleration due to gravity.
Elevator at rest or moving at constant velocity (inertial frame): The apparent weight is equal to the true weight.
N - mg = 0 \implies N = mg
Elevator accelerating upwards (\vec{a} upwards):
N - mg = ma \implies N = m(g+a) (Apparent weight is greater than true weight).
Elevator accelerating downwards (\vec{a} downwards):
N - mg = -ma \implies N = m(g-a) (Apparent weight is less than true weight).
Elevator in free fall (cable breaks, a = g downwards):
N - mg = m(-g) \implies N = 0 (Apparent weight is zero; feeling of weightlessness).
III. Equilibrium and Free-Body Diagrams
A. Free-Body Diagram (FBD)
A visual tool used to analyze forces acting on an object.
It shows the object (modeled as a particle) separated from its surroundings.
All external forces acting on the object are represented as vectors (arrows) with their appropriate magnitudes and directions.
Forces exerted by the object are not shown in its FBD.
B. First Condition of Equilibrium
Statement: An object is in equilibrium if the net force acting on it is zero.
Mathematical Expression:
\sum \vec{F}_{net} = 0
This implies that the sum of forces in each orthogonal direction must also be zero:
\sum F_x = 0
\sum F_y = 0
\sum F_z = 0
This condition is applied to determine unknown forces acting on a body modeled as a particle when it is at rest or moving at a constant velocity.
C. Accelerating System of Masses (Atwood Machine)
The Atwood machine was the first laboratory apparatus that provided experimental verification of Newton’s laws of motion.
Cables and ropes are efficient ways to transmit force within a system of masses.
For a simple Atwood machine with two masses (m1 and m2) connected by a rope over a pulley:
The acceleration (a) of the system can be derived from (m2 - m1)g = (m1 + m2)a
Thus, a = \frac{m2 - m1}{m1 + m2} g
IV. Friction
Definition: Friction is a force that resists motion (or impending motion) between surfaces in contact.
Friction exists in all types of materials.
A. Laws Governing Friction
Static vs. Kinetic Friction: Static friction is generally greater in magnitude than kinetic friction. This is why it requires more force to start an object moving than to keep it moving at a constant speed.
Direction: Friction acts parallel to the surfaces in contact and specifically in a direction that opposes the motion or the tendency of motion. Crucially, friction cannot cause motion; its role is always to oppose it.
Independence of Contact Area and Speed: Friction is largely independent of the apparent area of contact between the surfaces and the speed of sliding (for reasonable speeds).
Proportionality to Normal Force: Friction is directly proportional to the normal force (N) pressing the surfaces together.
Surface Characteristics: Friction depends significantly on the nature of the surfaces in contact and their condition (e.g., rough, smooth, polished, wet).
B. Types of Friction and Formulas
Static Friction (F_s):
The force that opposes the start of motion between surfaces at rest relative to each other.
It can vary from zero up to a maximum value.
Formula: Fs \leq \mus N
Maximum Static Friction (F_{s,max}): The maximum force that must be overcome to initiate motion.
Formula: F{s,max} = \mus N
\mu_s: Coefficient of static friction (dimensionless, depends on the surfaces).
Kinetic Friction (F_k):
The force that opposes motion between surfaces that are sliding past each other.
It is generally constant once motion has begun.
Formula: Fk = \muk N
\muk: Coefficient of kinetic friction (dimensionless, depends on the surfaces; usually \muk < \mu_s).
Rolling Friction: Friction that opposes the rolling motion of an object over a surface (generally much smaller than static or kinetic friction).
Fluid Friction (Drag Force): The resistive force exerted by a fluid on an object moving through it.
Relevant in the design of parachutes, where Louis-Sébastien Lenormand (1783) is credited with its invention, though Leonardo da Vinci (15th century) had earlier sketches.
C. Banking of Roads
Roads are often banked (tilted) on curves to allow vehicles to navigate them safely at higher speeds.
The banking angle leverages the normal force to provide the necessary centripetal force, reducing reliance on friction to prevent skidding.
Designed Speed Formula: For a frictionless banked curve, the designed speed (v) is given by:
\tan \theta = \frac{v^2}{rg} or v = \sqrt{rg \tan \theta}
Where \theta is the banking angle, r is the curve radius, and g is the acceleration due to gravity.
V. Newton's Law of Universal Gravitation and Kepler's Laws
A. Newton's Law of Universal Gravitation
Concept: Based on the observation that objects like an apple are attracted to Earth, Newton proposed that all objects in the Universe attract each other in a similar manner.
Statement: Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Formula: F = G \frac{m1 m2}{r^2}
F: Gravitational force between the two masses.
m1, m2: Masses of the two objects.
r: Distance between the centers of the two masses.
G: Universal Gravitational Constant, with a value of 6.674 \times 10^{-11} \text{ Nm}^2/\text{kg}^2.
B. Kepler's Laws on Planetary Motion
Developed by Johannes Kepler, these empirical laws describe the motion of planets around the Sun, later explained by Newton's law of universal gravitation.
Kepler's First Law: Law of Orbits
Statement: All planets move in elliptical orbits with the Sun at one focus.
This implies that any object bound to another by an inverse-square law force (like gravity) will move in an elliptical path.
The second focus of the ellipse is empty.
The eccentricity of an orbit is zero for a perfectly circular orbit and increases as the ellipse becomes more elongated.
Kepler's Second Law: Law of Areas
Statement: A line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time.
This means that planets move faster when they are closer to the Sun and slower when they are farther away.
For example, the area swept from point A to B will be the same as from C to D if the time taken is equal.
Kepler's Third Law: Law of Periods
Statement: The square of the orbital period (T) of any planet is directly proportional to the cube of the average distance (a) from the Sun to the planet (the semi-major axis of the elliptical orbit).
Formula: T^2 = K a^3
T: Orbital period of the planet.
a: Average distance from the Sun (or length of the semi-major axis).
K: A constant value that is the same for all objects orbiting the same central body.
For orbits around the Sun, K = K_S \approx 2.97 \times 10^{-19} \text{ s}^2/\text{m}^3.
The constant K is independent of the mass of the orbiting planet.
Relationship to Universal Gravitation: The constant KS can also be expressed in terms of the gravitational constant (G) and the mass of the Sun (MS):
KS = \frac{4\pi^2}{GMS}
This allows calculation of the Sun's mass if Earth's orbital period and distance are known: M_S = \frac{4\pi^2 a^3}{GT^2}. For Earth, T = 3.156 \times 10^7 \text{ s} and a = 1.496 \times 10^{11} \text{ m}.
VI. Practical Applications and Career Focus
Impulse-Momentum Theorem: Manny Pacquiao's powerful punches illustrate the impulse-momentum theorem (F_{avg} \Delta t = \Delta p), where a greater impact force results from delivering a given change in momentum over a shorter time or at a greater speed.
Aerospace Engineering: A career path for those interested in aerodynamics. Aerospace engineers design, construct, and test rockets and aircraft. They also research new materials and technologies. A bachelor’s degree in mechanical engineering is typically required before specializing in aerodynamics.
Geosynchronous Orbit: A special type of satellite orbit where the satellite remains at the same location relative to a point on Earth. This requires the satellite's orbital period to match Earth's rotation period (approximately 24 \text{ hours}). The distance (r) for such an orbit can be calculated using r = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3}, where M is the mass of the Earth.